Quantum Linear Algorithm for Edit Distance Using the Word QRAM Model
Abstract
Many problems that can be solved in quadratic time have bitparallel speedups with factor $w$, where $w$ is the computer word size. For example, edit distance of two strings of length $n$ can be solved in $O(n^2/w)$ time. In a reasonable classical model of computation, one can assume $w=\Theta(\log n)$. There are conditional lower bounds for such problems stating that speedups with factor $n^\epsilon$ for any $\epsilon>0$ would lead to breakthroughs in complexity theory. However, these conditional lower bounds do not cover quantum models of computing. Indeed, Boroujeni et al. (J. ACM, 2021) showed that edit distance can be approximated within a factor $3$ in subquadratic time $O(n^{1.81})$ using quantum computing. They also showed that, in their chosen model of quantum computing, the approximation factor cannot be improved using subquadractic time. To break through the aforementioned classical conditional lower bounds and this latest quantum lower bound, we enrich the model of computation with a quantum random access memory (QRAM), obtaining what we call the word QRAM model. Under this model, we show how to convert the bitparallelism of quadratic time solvable problems into quantum algorithms that attain speedups with factor $n$. The technique we use is simple and general enough to apply to many bitparallel algorithms that use Boolean logics and bitshifts. To apply it to edit distance, we first show that the famous $O(n^2/w)$ time bitparallel algorithm of Myers (J. ACM, 1999) can be adjusted to work without arithmetic + operations. As a direct consequence of applying our technique to this variant, we obtain linear time edit distance algorithm under the word QRAM model for constant alphabet. We give further results on a restricted variant of the word QRAM model to give more insights to the limits of the model.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.13005
 arXiv:
 arXiv:2112.13005
 Bibcode:
 2021arXiv211213005E
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 81P68;
 E.1;
 E.4;
 F.1.3;
 F.2.2
 EPrint:
 An incorrect assumption invalidates the results